In recent years, due to the sustainability concerns, energy-efficiency programs that address eco-friendly issues such as energy efficiency and clean production have increased in the world. Furthermore, eco-efficient appliances, as the direct result of research and development (R&D) activities have become a popular topic in the sustainability programs of the most developed countries due to their high-tech features. In this paper, we discuss a residential energy-efficiency program for the new energy-efficient appliances and related energy consumption, simultaneously, in a sustainable supply chain consists of the energy supplier, manufacturer, and the government. Also, we consider the technological rebound effect of the efficiency improvement, consumer behavior, and ability to innovate of the manufacturer in the problem, as the novel assumptions. To solve the problem, we propose a multi-stage game model using the Bertrand model. Also, we investigate three structures, based on the Nash and leader-follower games as the scenarios. The findings reveal the equilibrium pricing decisions for the supply-chain players to maximize their profits, besides the best energy policy and supply-chain structure for the efficient management of household energy consumption. More precisely, the first scenario with the Nash structure between the manufacturer and energy supplier has the most advantages and the least disadvantages than the other scenarios. Also, the manufacturer subsidy has the most performance in the discussed energy-efficiency program. Finally, the present study shows the significant effects of considering the energy rebound, innovation abilities of the manufacturer, and consumer behavior assumptions in the household energy problems.